An example of elliptic stability with large parameters: Lamé’s equation and the Arnold-Moser-Rüssmann criterion
Authors:
G. Pecelli and E. S. Thomas
Journal:
Quart. Appl. Math. 36 (1978), 129-140
MSC:
Primary 70.34; Secondary 34D20
DOI:
https://doi.org/10.1090/qam/502471
MathSciNet review:
502471
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F. M. Arscott, Periodic differential equations, Pergamon, 1964
- Richard Churchill, Henry Jurenka, and Edward S. Thomas Jr., A note on isoenergetic stability, Internat. J. Non-Linear Mech. 12 (1977), no. 1, 37–43. MR 449082, DOI https://doi.org/10.1016/0020-7462%2877%2990015-4
- R. C. Churchill, G. Pecelli, and D. L. Rod, Hyperbolic structures in Hamiltonian systems, Rocky Mountain J. Math. 7 (1977), no. 3, 439–444. MR 467817, DOI https://doi.org/10.1216/RMJ-1977-7-3-439
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- Thomas G. Proctor and Raimond A. Struble, Motion of two weakly coupled nonlinear oscillators, Arch. Rational Mech. Anal. 18 (1965), 293–303. MR 171974, DOI https://doi.org/10.1007/BF00251668
- R. M. Rosenberg and C. P. Atkinson, On the natural modes and their stability in nonlinear two-degree-of-freedom systems, J. Appl. Mech. 26 (1959), 377–385. MR 0108009
- Helmut Rüssmann, Über die Normalform analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann. 169 (1967), 55–72 (German). MR 213679, DOI https://doi.org/10.1007/BF01399531
- E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469
V. I. Arnold, Kleine Nenner und Stabilitätsprobleme der klassischen Mechanik und der Himmelsmechanik, Uspekhi Mat. Nauk USSR 18, 6, 9–192 (1963)
F. M. Arscott, Periodic differential equations, Pergamon, 1964
R. Churchill, H. Jurenka and E. Thomas, A note on iso-energetic stability, Int. J. Non-linear Mech. 12, 37–43 (1977)
R. Churchill, G. Pecelli, D. Rod, and S. Sakolic, Coexistence of stable and random motion, Rocky Mountain J. Math. 7, 445–456 (1977).
N. W. McLachlan, Ordinary non-linear differential equations in engineering and physical sciences, 2nd edition, Clarendon Press, 1956
W. Magnus and S. Winkler, Hill’s equation, Interscience Publishers, 1966
J. K. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1 (1962)
J. G. Proctor and R. A. Struble, Motion of two weakly coupled nonlinear oscillators, Arch. Rat. Mech. Anal. 18, 293–303 (1965)
R. M. Rosenberg and C. P. Atkinson, On the natural modes and their stability in non-linear two-degree-of-freedom systems, J. Appl. Mech. 26, 377–385 (1959)
H. Rüssmann, Über die Normalform analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Annalen 169, 55–72 (1967)
E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th edition, Cambridge University Press, 1952
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Article copyright:
© Copyright 1978
American Mathematical Society
